How many patterns can you create with a thread while sewing on a four-hole button?
Unfortunately, there is no information on the Web what the answer is and had Kolmogorov found it himself. In a book about Grigori Perelman, its author Masha Gessen admits that two professional mathematicians, both students of Kolmogorov, have given different answers. I am not a mathematician, but think one solution to the problem is following.
A1. It is mandatory to utilize the holes. There are many ways to sew on a button without utilizing the holes but let us forget about them for the time being.
A2. All holes are located on a straight line.
1. Let us take a button with n holes (n is a natural number equal to or greater than 2). As all holes are located on a straight line, there are n-1 segments between the outermost holes.
2. Let us call a segment black if a thread connects the holes and blank if there is no connecting thread. The total imaginable number of patternsis 2^(n-1), where 2 is the number of colors (black and blank).
3. The total actual number of patterns f(n) is [2^(n-1)]-1, as we should not count the case when the color of all segments is blank (in which case the button is not attached at all).
In the case of Kolmogorov’s button f(4)=7.
For those who disagree with the assumption that button’s holes are located on a straight line (except for the case of n=2) a separate text is coming soon.